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Course: Precalculus > Unit 7
Lesson 6: Using matrices to manipulate dataUsing matrices to manipulate data: Game show
In other videos, we saw how we can use matrices to represent real-world situations. Once we do that, we can also manipulate the matrices (using common matrix operations like addition, subtraction, and scalar multiplication) to reveal more information about the context. Here, we manipulate matrices that represent the possible prizes in a game show. Created by Sal Khan.
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"We're told, at the beginning of each episode of a certain game show, each contestant picks a certain door out of three doors. Then the game show host randomly picks one of the two prize bundles. After each round, each contestant receives a prize based on the door they picked and the bundle the host picked."
Why is this so confusing?(2 votes)
The statement may be confusing because it contains multiple variables and actions that occur in a specific order. Additionally, it may be unclear what the specific rules or mechanics of the game show are. However, with a bit of attention and re-reading, it is possible to understand the scenario. Essentially, each contestant selects a door, and the game show host selects a prize bundle. The prize the contestant receives depends on which door they selected and which prize bundle the host picked. There are also multiple rounds, and in the second round, the prizes are doubled if it is a lightning round.(2 votes)
I found it a little confusing whether the lightning round would be doubling Matrix A or Matrix B. This didn't seem clear from the question.(1 vote)
It seems that the values for D only accounts for the amount of money you can get by when the contestent choose the same door and the host picked the same bundle for both rounds - so the information it provides is fairly limited. Is it that I've mistaken? Because Sal didn't mention this.(1 vote)
Video transcript
- [Instructor] We're
told, in the beginning of each episode of a certain game show, each contestant picks a certain
door out of three doors. Then the game show host randomly picks one of the two prize bundles. After each round, each
contestant receives a prize based on the door they picked and the bundle the host picked. Matrix A represents the possible prizes for the first round. All right, so for example, if the contestant picks door three and the host picks bundle
one, the prize is $300, but if the contestant picks door three and the host picks bundle two, the prize would be $0, all right. And then they say matrix B
represents the possible prizes for the second round. All right, that's fair. They also tell us the
second round can also be a lightning round. In this case, the prizes are doubled. Matrix C represents the possible prizes during a lightning round. Complete matrix C. So pause this video and see
if you can figure that out and then we'll work through this together. All right, so matrix C is a
scenario where we're dealing with a lightning round, and remember in a lightning
round, the prizes are doubled but it's the second round. It's doubled relative to
what it would have been in the second round. So what it would have been in
the second round is matrix B. So another way to think about it is, matrix C is going to be
equal to two times matrix B. And we know when we multiply a matrix times a scalar like this,
times just a number, we just multiply each of
these entries by that number. So let's do that. If we take $600 and multiply that by two, that is going to be $1,200. And that makes sense. We just said for each
corresponding scenario the prizes are doubled
in a lightning round. So if the contestant picks door one, host picks bundle one, instead of $600 it's going to be $1,200. Keep going, instead of
$200, it's going to be $400. All I'm doing here is I'm
multiplying each of these entries by two to get the corresponding
entry in matrix C. Keep going, instead of $300 here, multiply that by two,
you're going to get $600. Instead of $300 here,
you're going to get $600. We're almost there. Instead of $0 here, well, zero times two is still $0. And then last but not least, instead of $400 right over here, that times two is going to be $800. Now there's one more question that they have below the
screen right over here. Let me scroll up a little bit. So they tell us matrix
D is defined as follows: D is equal to A plus B. What does matrix D represent? So pause the video and think
about that for a second. Well, if we add two matrices, we're going to add all
the corresponding entries. And so what it tells you is
what is the combined prize for both rounds one and two based on what the contestant
picks and what the host picks. So matrix D, that top
left entry will tell you, okay, in total, if the
contestant picked door number one and the host pick bundle
number one, what would you get? Cause it would be $100 dollars plus $600. So it would be total
for rounds one and two, assuming we don't have the lightning round like we had in matrix C.